


Superheroes: Powers and Principalities

by Charles_Rockafellor



Category: Meta - Fandom
Genre: Anime, Chi - Freeform, Kardashёv, Magic, Power scale, Psionics, RPG, Superpowers, Superscience, Writing, ZFC, Zermelo-Fraenkel, gch
Language: English
Status: Completed
Published: 2021-02-14
Updated: 2021-02-14
Packaged: 2021-03-18 16:13:31
Rating: General Audiences
Warnings: No Archive Warnings Apply
Chapters: 1
Words: 7,689
Publisher: archiveofourown.org
Story URL: https://archiveofourown.org/works/29371374
Author URL: https://archiveofourown.org/users/Charles_Rockafellor/pseuds/Charles_Rockafellor
Summary: This is an examination of how to scale power levels of characters (comics, anime, movies, writing, RPGs, etc.).  Specifically, it acts as a general guide for small differences of rank (e.g.: John Q. Public might be rank D, and Batman Rank A) and exponential differences of tier (e.g.:  John Q. Public and Batman are both human, nothingoverlyspecial, Tier 1; the Justice League or X-men Tier 2; Q of ST:TNG likely Tier 3...).  It's a meta-work.Being into mathematics isn't necessary here, though fellow math geeks might have a ball with Knuth's up-arrow notation (and a little ZF(C)).𝑫𝒐𝒏'𝒕 𝒇𝒐𝒓𝒈𝒆𝒕 𝒕𝒐 𝑳𝒊𝒌𝒆, 𝑺𝒉𝒂𝒓𝒆, 𝒂𝒏𝒅 𝑺𝒖𝒃𝒔𝒄𝒓𝒊𝒃𝒆! ❤️
Collections: Worldbuilding Meta





	Superheroes: Powers and Principalities

**Author's Note:**

> At some point between early June 2019 and late July 2019, I looked at all of the characters that I had written into my Icewall stories (I didn't join AO3 'til much later) and decided that they needed organizing. In super-hero RPGs, you generally see them sorted out in some obvious power hierarchy -- and DC Heroes certainly did a good job of making it mathematically sensible. So I looked at my (N)PCs and scratched my head: how best to group such disparate stat-scores from the Ants (of “The Colony”, currently just an outline with worldbuilding [on my Google drive](https://drive.google.com/drive/folders/1BG7P_sVezz8Dn5b8js_34yeAjSBhfN0v)) through Negent (of “[I am Legion](https://archiveofourown.org/works/24354049/chapters/58727857)”)? Since Negent prompted all of this, it was the “I am Legion” .pdf that I stuck the results into (in what is currently “Appendix F: Powers and Principalities”, whence this article’s title).
> 
> The direction that I took with it started from the sensible base 10 scaling of “DC Heroes” RPG (much though I loved Marvel's MSH RPG and Palladium's Heroes Unlimited) and Traveller RPG’s tech particulars, and applied them to arbitrary extensions of the Kardashёv scale (or “Kardashev”, for the search engine).
> 
> Examples given cover a number of different genres: use whatever works for you, and modify as needed for your setting; ditch the rest.
> 
> For further meta reading, see also:
> 
>   * “[The Hadron Octahedron](https://archiveofourown.org/works/29512119)”, 
>   * “[GEN 0 particles](https://archiveofourown.org/works/29688639)”, 
>   * and “[The Categorical Logic Cube](https://archiveofourown.org/works/29874783)”. 
> 

> 
> **N.B.:** Contains some spoilers to some of my stories. Real-world fiction spoilers are marked clearly.
> 
> * * *

  
**Contents:**  
[Intro.]  
Zeroes  
Ranks F-A  
Tier 2 Supers  
Moorcock entities  
Dreamtime  
Beyond  
**TL;DR**  
[NB]  


#### Quick intro.

In the words “Power” and “Principality”, I'm not consciously cribbing notes from Tolkien or Judaeo-Christian teaching, nor quite from the political arena, but they're all influences subconsciously. 

(N)PCs scale enormously in power here, but the scaling isn't easy to define, which is made the worse by the syncretic structure of their functions. Prima facie... it's like going from fine-tuned differences of just how many fingers you have available to count on to the [comparisons/contrasts](https://wiki.travellerrpg.com/Tech_Level_Comparison_Chart) of [Traveller and 3e/4e GURPS tech levels'](https://wiki.travellerrpg.com/Technology_Level) differences (and types), and then on to [Kardashёv](https://en.wikipedia.org/wiki/Kardashev_scale) types (and as with the movie “ _Cast away_ ”, one might possess advanced knowledge and/or power, but be unable to do anything technological or otherwise with it, which is the case in some D&D worlds with low background magic-energy resources [see 1e & 3e “ _Manual of the planes_ ”] and throughout The Ice [world-ponds and rare oases aside]: an electrical lamp is a paperweight when the electricity is out). 

Speaking of The Ice: when I wrote this, it was targeted at my own story/game-world of “Icewall”, but that doesn't affect the thrust of scaling characters' powers. Just treat those references as being a setting that you're not familiar with (basically one where a weird Bermuda Triangle-like effect suppresses tech at flashlight-level complexity or above) and move along to the generalities or the next point (though if the idea sounds interesting, you can find the Icewall world-book [.pdf on my Google drive](https://drive.google.com/drive/folders/1BG7P_sVezz8Dn5b8js_34yeAjSBhfN0v)). 

I'll stick to the general anime/manga ranking method for this one: assume ranks (F), D, C, B, A, S. You can add AA / AAA or A+ to this, but the principle remains the same. 

* * *

####  Zeroeth tier: non-ranked **Get to the point ▼ Contents ▲**

Life of any kind, or even just the very existence of non-living material, is miraculous – grok some chemistry or physics, or meditate (religiously or philosophically), and you'll be amazed – but let's face it: your average paramecium isn't exactly operating on the same level as your average house cat or human. This means that there are clearly levels that would fall far short of typical storybook or game characters' capabilities (whether average, or super-powered, or challenged by innumerable slings and arrows). 

For argument's sake, I'll simply place them at Kardashёv Type ε (in the sense of (1/ℵα = **:** εα) > 0 and [watch the smoke and mirrors behind the curtain] εα2 ≝ 0, rather than in the sense of (ε0 **:** = ω↑↑ω) ≫ ℵn). 

This isn't an accurate rating, just evocative. Technically, the Kardashёv civilization typing is a logarithmic scale, so paramecia and such might be Type -1 or -2 or something, but I'm going to pull artistic liberty here and just proclaim them to be ε by fiat in order to drive home the relative insignificance of their power levels, knowledge base, etc. (“-1 or -2” just doesn't feel as if it has quite the same impact as ε). 

#####  0: Micros' takeaway points 

They exist, they use tooth and claw (or some equivalent). That's about it. 

[≪10e6 Wh/d (i.e.: ≪K 0)](https://hypertextbook.com/facts/2003/WeiLiangMok.shtml) (where 10e-4 W = K -1). 

In this sense, consider K ε as representing a power/etc. level that's asymptotically close to |0|, rather than the more accurate equivalence of K 0+. 

Effectively what would be _Classic_ Traveller TL ε (with same clause as K ε). 

* * *

####  First tier: ranks F-A **Get to the point ▼ Contents ▲**

Your average human being, or at least the range that one might expect people to fall into. A small child might struggle with a 10 pound weight, and adult can typically lift and carry 100 pounds (within a factor or so, more or less), and heavy-weight lifters might well reach 1,000 pounds (squat, bench press, or dead lift; perhaps half of this for snatch, or clean and jerk). 

The average Ants in “ _The Colony_ ” would rate their own special Rank F, they're _that_ low-statted – but hey, they're better off than the (N)PCs of _Creeks and Crawdads_ (I can't tell if the game .pdf [or the _Never cry Crawdad!_ module .pdf] is legitimately free or not [sorry: I don't wish to risk copyright problems, so you'll have to search for those yourself], so I'll link only the definitely-free [Impact System revision](https://www.fictioneers.net/games/creeks-and-crawdads)), who'd legitimately earn a Rank F-. <sigh> Those poor Ants and Crawdads. 

The Octorillas and Cuccos of “ _[Sonic's Redemption](https://archiveofourown.org/works/24230851/chapters/58380841)_ ” would qualify for this tier (though not all cuccos are Cuccos **1** ). The Octorillas are sophont, though only slightly sapient; the Cuccos are somewhat more sapient (their corrals and husbandry techniques showing some sophistication as tools in their own right), though only somewhere between 0 and 1 in sophonce. Both are fully sentient, but neither has any super powers. I suppose put them each as F+ (even though the Crawdads might, and the Ants apparently do, have a slightly more advanced overall physical setting and clear social interaction than the Octorillas and Cuccos have). 

Your average peasant, John Q. Public, etc. – such as TMNT's Casey Jones (if I remember correctly, anyway) – is Rank D: not remarkably good at anything, not remarkably bad at anything. 

Friday (a Deku Scrub of “ _Sonic's Redemption_ ”) has mystical knowledge (though I'm not sure of just how much), and has applied it shamanically (at least pharmacologically, and possibly spiritually/Jungian archetypally), but hasn't yet demonstrated any actual magical powers per se, hence is at least Rank C (possibly higher, of course, just not lower). 

Deathclaw (Mr. Fluffykins, of “ _[Deathclaw: the tale of Mister Fluffykins](https://archiveofourown.org/works/24352675)_ ”) is probably Rank C in raw power and fighting, or just maybe Rank B (by now, at least – he was at the end of his rope when he met Peach, and likely ranked as an overall D or F at the time). This is in an absolute sense (using the F...A scale relative to typical humans), since his current Awakened state would qualify him as Rank S relative to other cats, based upon his ability to speak and read Common. Some cats can use doorknobs and toilets (I had one who'd learned both, and another who'd figured out only the toilet, but each of them was completely self-taught); while such skills are fairly remarkable (given the low rate of their occurrence), they aren't S-rank level. 

Robin Hood, though an older reference, would likely qualify as Rank A (or a strong B at the very least). 

Batman is Rank A (or arguably A+, for a finer-tuned distinction). 

None is superhuman. 

The overall civilization of _Creeks and Crawdads_ probably stands at about the level of your average kindergarten class... minus the teacher. 

The Ants' civilization is operating at perhaps 1st (just maybe 2nd) grade level... also minus the teacher (but worse: the Ants are being run by someone who might as well be Angelica, of _Rugrats_ ). 

<sigh> Those poor Ants and Crawdads. 

#####  1: F-A takeaway points 

People have cars and computers, or something of an equivalent level; ~30 kWh/d. Nukes and such (or equivalent) exist; civ. ~10e13+ bits data & ~10e10-10e11+ kWh/d ≈ 10e13-10e17 W/y (these being broad orders of magnitude and using rather varying sources of information about power demands). 

Individually, people are estimated at a 100 W throughput (10e2 W → K -0.4 ≈ K 0); by daily use of items at our command (though pretty much only “first world” usage) 30 kW ~K 0. 

All stand at around Kardashёv Type 0. 

Individuals might be trained in skills of any Classic TL, but won't have any innate powers as such; an everyday citizen could be expected to have Classic TL 6-9 (depending upon the type of tech. in question) at their immediate disposal. Their civilization could reach Classic TL 8-10+. Clearly, these assume something like 20c-21c Earth, and would presumably be higher or lower at other points in time. 

* * *

####  Second tier: S-rank Supers **Get to the point ▼ Contents ▲**

Your average superhuman.

Superman, Thor, Wonder Woman, Dr. Strange, Hulk, Spider-Man, Dark Phoenix, Hercules, and anyone else who can do things – even a single something (even if just _barely_ ) – that absolutely positively **_can't_** be performed by an unaided, unaltered, non-mutated, non-magical, non-psionic, non-cybernetic, non-whatever-else ordinary everyday human, no matter how incredibly buffed up that human's stat.s are, is at least Rank S (or possibly on the teetering edge of S-, if you somehow need such a thing). No, that doesn't mean that birds and fish are Supers. 

How many “S” ranks are there? I mean, not all cybernetic-super-mutant-mages are born equal. Doesn't matter specifically, but let's say that there are at least 3 (I'm comfortable with 5, too, but that's irrelevant). I don't know whom you care to rank where, but Doctor Manhattan (perhaps?), Q, and The Beyonder (and Dr. Doom, for a little while there...) are really way out in a class of their own: Rank SSS or Rank SSSSS, given the ranking system _so far_. 

In order to keep it easy to read, I'll use subscripts here: S3 or Rank S5. 

A normal human is roughly equivalent in _level_ to the state that they're shown as being in “ _Fantastic planet_ ” (or as described in “ _Battlefield Earth_ ” [the pulp sci-fi book's description, not the craptastic movie depiction])...: kitty-pets (or verminous shitty-pests, depending – particularly if viewed from something like the perspective that one might imagine of Titans in “ _Shingeki no Kyojin_ ”), essentially; not precisely far- _far_ beneath the level of power and intellect of a Super (hold onto your objection 'til after you've read tier three for contrast), but not exactly on an even footing either. 

Strictly, a zombie wouldn't fall into the F...A tier since they're somehow defying death if they're undead (a real zombie then – not simply a rage-zombie that's a living being with some virus). On the other hand, if that's the _only_ condition that applies (and the attendant facts that they can certainly take a world of punishment from environmental to starvation and keep on ticking), then they aren't very far from normal people either (call them... “living impaired, plus perq.s”). I'll call them S-, but I couldn't really fault anyone for dropping them to something like A+ (this wouldn't be accurate to me, given the details here and the ranking method so far, but... ehh). 

Vampires, Werewolves, Wendigo (the usual type, not the variation that Warbotdorf [a later alias of Robotnik] refers to in “ _Seize the Deity_ ”)... I'll skip the rant about monsters with human faces (narcissists); the parallels to _World of Darkness's_ Wyrm's draining abuse, all-consuming rage, and mind-blotting need to consume and destroy are apt but off-topic. Taken as metaphors, vampires and such are still just humans, hence F...A; treated as literal super-powerful beings within a game or a story though, they're S1 or higher (usually not higher, but it could happen: consider an antediluvian or even just a methuselah of VtM, or Akasha of Anne Rice's “ _Queen of the Damned_ ” – hell, consider Imhotep of [the 1999 version of] “ _The Mummy_ ” [the good one, with Brendan Fraser]). 

The city of Ankh-Morpork (Terry Pratchett's _Discworld_ series) could qualify as merely Tier 1 (F...A), or even merely 0 (:: paramecia), though I can't rule out Tier 2 (Supers and Powers), based upon >>> **_SPOILER ALERT_** >>> the events of “ _Reaper Man_ ” (the book with the snow globes and the shopping carts). <<< **_END OF SPOILER_** <<<

The seemingly-living corporate embodiments in the movie >>> **_SPOILER ALERT_** >>> “ _Branded_ ” (you might recall something from it about a red heifer?) <<< **_END OF SPOILER_** <<< possibly qualify for S1 or S2, in some ill-definable paranormal sense. 

WHFRPG Gods and some D&D Gods would probably(?) be Rank S1, or Rank S2 at most. 

If we assume 5 ranks in the S tier, then Superman might be S3 or higher (for the general degree of each power, not the scope of his powers). 

I'm less familiar with Wonder Woman, but her divinely-gifted powers are [greater than his, by canon](https://wonder-woman.fandom.com/wiki/Wonder_Woman)). Whether they're sufficiently greater to place her up a rank I leave to your discretion. 

The Fylgjur of “ _Nyjötnar Saga_ ” are a bit iffy (I'm only as far as the barest outline and some worldbuilding): I haven't really developed that story very much yet, so I don't have a good handle on all of the details, but if their Fylgjur more or less approximate the norm, then they have some capability beyond everyday people's, but not really on the level of a comic book character. Let's give them a placeholder of S- for now. 

The Ant Queen of “ _The Colony_ ” I'll place as S1 by way of her ability to pump out gene-tailored kids. Moreover, they're not simply GMO, but can include flight and fire-breathing (dragons that are still in paper-note form, awaiting transcription), T-virus-like hulking brutes, and a few other mutant superpowers. Yeah, I'd say that this counts. 

The Augments and Cybers of “ _Sonic's Redemption_ ” would fit in here as Tier 2 Supers – as long as you'd consider the Six Million Dollar Man or Daleks as being something along the lines of super powered characters. 

Kaonashi might qualify for Rank S1 (footnote in Afterword of .pdf of “ _Seize the Deity_ ”), though I'm not so sure of Caïssa (footnote in Epilogue of .pdf of “ _Seize the Deity_ ”), as would the Great Deku Tree that I never name explicitly (it's world-building background in the back of my head) as being the source of protection (illusory terrain, ghost sound, various vermin and predatory plants throughout the outskirts) that keeps the Deku Scrubs' ancestral lands safe from exploitation. 

Other beings – such as Icewall-Titans (something like Dune mélange-induced Greek gods or ST:TOS Apollo / Charlie X) or Icewall-Cthulhu, and some D&D Gods – would fall someplace around Rank S3...5. 

Icewall's world-trees that seed the plane with mountainous seeds shot like volcanic cousins to the [movie/TV-version] Klendathu (of “ _Starship troopers_ ”) are K1 in power, though not canonically named as being intelligent or not (perhaps a huge store of information genetically, but with a K0- consciousness {sapience, sentience, sophonce} on par with what a character might normally expect of a worm or a tree). Though their [lack of] volitional awareness wouldn't rank them as a Power, the sheer scope of their natural abilities (even if only in a general ecological niche fashion) places them here in the same sense as worms, house cats, and non-super humans are all in the same ballpark in contrast to prokaryotes or Supers. 

Each additional S rank leaves the prior ones in the dust to the extent that each F...A rank does the same within the F...A tier. Clearly though, there's a huge difference between the F...A tier and the S tier. 

Zelda's a Goddess (herein an autonomous avatar) and [in my story-world] Peach is Ōkami. They're Powers. That puts them someplace in the S tier (I think...?), and I'd guess Rank S2 \- Rank S3, since I just don't see them as falling into “merely” Rank S1. 

Their incarnate powers don't seem to be much to speak of, but we know that the canon of the real [video game] Hyrule shows Din, Farore, and Nayru as having created that world. That's some reasonably vast power compared to your average superhero. Hylia might have the same level or not, I don't know (could be higher than theirs, could be lower). 

We've seen Peach pull off her little auto-reincarnation feat within this story-verse, and discussed some of the implications thereof regarding disincarnated Ōkami powers (see “ _Seize the Deity_ ” .pdf ► DELETED scene ► footnote). Those are some serious abilities. 

Sonic and Tails – neither of whom will stay dead, short of a nova – pretty much qualify for S tier. I'd say Rank S1, but presumably(?) not Rank S2. Tails's current INT is S1 on its own, and might combine with everything else to boost him to S2. 

Ruru's gotta be at least Rank S-, simply based upon her spell-like abilities. (Yeah, this technically opens the question of _everyone_ with _any_ magic, psionics, chi, etc. as qualifying for S tier, but let's skip the pedantry; for now, just assume that alchemists and wizards learn things and hence don't qualify for more than A+ as-is, but also that _how_ and _how much_ wizards do what they do is beyond human bodies' natural capabilities just as much as sorcerers' innate control, and thus qualify them as S- or better – a warlock's pact-granted-powers would be hand-me-down sorcery on par with Doc Oc's cyber-powered super-strength (or Spidey's, since that came from a bite rather than au naturel [and let's face it: Iron Man is just Tony in a flying tank, but it still sort of fits into the S ranks]). A spell-like ability that simply does otherwise normal F...A things would qualify as S1 in its magical nature and complete lack of necessity for the normal human approach, even if not in its scope of being merely an F...A result. 

Amy too (Amy Wong, the AI of “ _[I am Legion](https://archiveofourown.org/works/24354049/chapters/58727857)_ ”), by dint of being essentially a liquid metal Terminator: S- to S1. Her INT skews things a bit, since I suspect it of being at least on par with Tails's, which might bring her to a total of S1...S2.  
Come to that though, Amelia (the other Amy, of “ _[In for an Amy, in for a Pond](https://archiveofourown.org/works/27330109/chapters/66774238)_ ”) would probably qualify as well, by being a Time Lady. 

Peach specifically stated (obliquely) in the epilogue of “ _Seize the Deity_ ” that Shīfù Lǎorén, the Immortal Wombat, is a Power – and she wasn't guessing or being polite. We know [from the Afterword in the .pdf] that he's immortal (can be killed, but won't simply die on his own), and that he's of an insanely high OCC-level (a 15 levels each, of everything combined into one style, ~465 per _Ninjas & Superspies_, but with an actual canonic effective level of about 1,200). Yeah: he's some kind of S tier, _minimum_ , whether literally or de facto super. 

Daemon (the _ReBoot_ character whom I used for “ _I am Legion_ ”) I'm not sure of – she might be multi-S, and she might be next-level (hell, bookmark her for now as being Sω). 

The Avatars in _Charmed_ (original 1998 TV series) are probably at her level. 

The Gaian of “ _Sonic's Redemption_ ” is somewhat spread across the tiers.

  * In terms of power that it can muster, or intellectual capability, it's pretty much dead in the water: tier 0 or tier 1; it's arguably on par with a sequoia or armillaria ostoyae (other than that its components _can_ move). 

  * Given its integration of organisms over unnamed vast distances to form its being, that's a super power of some kind: tier 2. 

  * In light of its purely zooic nature, it might reasonably fall into the bottommost ecological niche of tier 3: Principalities (see next tier-entry). 




“It” (i.e.: Morgoth, of “ _[When the paladins come marching in](https://archiveofourown.org/works/24371467/chapters/58778377%22)_ ”) probably lies somewhere within the above-outlined Sn range, but I don't have a bead on just where. 

All are superhuman in one way or another, and/or have access to laws of nature other than those that are commonly perceived (e.g.: they can perform “supernatural”-seeming feats through magic, psionics, chi), but none is _truly supernatural_ in the sense of _decreeing_ new laws of nature as such. At truly staggering ranks, the wishes, loyalties, and hatreds of F...A ranks are as irrelevant to them as an ant's agreement or objection might be to your drinking a glass of water or going shopping (note that this doesn't mean that a superhero wouldn't bind themselves to non-Super law, only that they _choose_ to do so by their own honor, whereas a super-villain feels no more bound to it than a non-Super villain, and is less constrained to do so). 

One could argue that modern Earth nations stand at this level, or at least at the very bottom threshold thereof. 

#####  2: Supers' takeaway points 

People run around with über-tech coursing through them as much as we've had mitochondria for billions of years and give it no thought. Tailored worlds and walk-through stargates exist; civ. ~10e13-10e15 bits data & ~10e16-10e26 W/y. 

Kardashёv Type ~I-II? (Clearly not invariably, since our heroes usually don't exhibit full K1 power levels, so take these with some latitude as an approximate upper limit.) 

Individuals might reasonably possess personal powers (GMO/mutant, psionic, nano, etc.) on the order of Classic TL 9-20 (variance due to type of tech. in question), whether typical or atypical of their civilization; an everyday citizen probably would have fairly unquestioned access to Classic TL 10-17 (depending upon the type of tech. in question). Their civilization overall could reach Classic TL 25+. 

* * *

####  Third tier: next-level incomprehensible **Get to the point ▼ Contents ▲**

Your average god-like entity, and I don't mean Greek Gods, I'm thinking cosmic-level shit here.

What about beings on the scale of [Moorcock's gods of Law and Chaos](https://en.wikipedia.org/wiki/Deities_in_the_Elric_series), since they're more or less my conception of Negent (and similar entities) – except that they seemed to be a bit limited in their potentia in contrast to what Negent is (then again, maybe they were more interactive than he, and operated under essentially the same constraints as he, hence it's simply more apparent with them)? They're clearly not F...A, but I wouldn't class them in the S tier and call it a day. The same might(?) be said for Discworld's Azrael or Adventure Time's Prismo (and others): these entities don't show their hands, so I can't really say shit, but I'm not sure that they're operating at anything less than T3 R3 (and honestly, I can't rule out T4). 

Maybe we should simply assume direct [hyperoperation](https://en.wikipedia.org/wiki/Hyperoperation#Examples) as the key at this point (e.g.: humans being additively more powerful, Supers multiplicatively so, and Principalities exponential), rather than which value is being hyperoperated on. This will get a little ad hoc, so please bear with me. 

A Principality (Void, Kallistṓ, Entropy, and Negent - all of “ _I am Legion_ ”) would qualify for the Sn tier (at that point, does the exact value of the |n|-Rank really matter all that much, in overall scale?). Technically, the S1, S2, etc. ranks fall within the Sn tier, in that they'd all be _factors_ of S1 – and by that same token, F...A ranks would be infinitesimal fractions of S1. Assume here that any significant Sn refers to _magnitudes_ of S (i.e.: n>1) – much like the differences between a point and a line segment, and an area, and a volume (though only somewhat, since a figure of n-dimensions [regardless of measure] always has the same number of DIM 0 points, as far as I know [ZFC **2** aside, anyway]) – such that: 

{S0 < S} :: {F . . . A}

{S1 = S} ≡ {S1 = 1*S = S};  
here S1 ≡ S1 (if one takes the lowers bounds as being subsumed),  
but more generally S1 ≫ S1 (where I take each tier as the next stairstep) 

{S2 :: S*S} > {S2 :: 2*S = S+S} 

{S3 :: S*S*S} ≫ {S3 :: 3*S = S+S+S} 

etc. 

If we do this, then the Principalities can be ranked roughly like superheroes and average people. 

If we further assume either the actual layout above or one extremely similar, then Negent and the others would probably be something like Rank S1...5. There's always the chance that there are other, hidden Principalities (such as one that arranges filigreed patterns, rather than emptiness, randomness, blended equilibrium, or polarizations – assuming that this weren't a subtle play by Tào, and that Tào isn't simply a Principality of great skill and delicacy), and their power could go off-scale, but for now we'll simply assume that at least these four are fairly closely balanced. Perhaps there are huge asymmetries, but only in dissimilar ways, much as with S tier: Brainiac's intelligence vs. Dormammu's magic vs. Saitama's punch – each is vastly qualified within their field of expertise, but how does one compare and contrast between them? Q and The Beyonder might (just barely) fall into this tier, rather than the standard S tier (Dr. Manhattan not so much, but this same “ _not so much_ ” arguably applies perfectly to Principalities, so... yeah). 

Primordium, named in the prologue of “ _[What evil lurks](https://archiveofourown.org/works/24360739/chapters/58747942)_ ” (Link's sidequel adventures to my main Zelda series) might fall into any tier above F...A. 

Whether it's a Power, a Principality, or something wholly other on even footing with Tào (“ _I am Legion_ ”) or something, I don't know, but it's definitely capable beyond merely human levels. I described it as chaos that strode the lands in darkness (with the deliberate lack of narrative cohesiveness regarding Primordium's logical place as an actor in history, given the inherent self-contradictory nature of Icewall), a thing which in turn found [life / goodness / something referred to indirectly] to be a brittle burr and consumed itself with a non-specific intense yearning [to destroy utterly this brittle burr], apparently capable of “... _stoking the currents that flowed across eternity, plucking the threads that guided all things, searching for the weak spot to harry_...”. The reader is [meant to be] led to believe [correctly] that it proceeds to do so through indirect means, not by clashing headlong with existence itself or any god-like beings, but by subverting life into malformed creatures of chaos and evil (active malevolence, rather than “mere” callous indifference) – such as Kandarian Demons (and presumably beings of the Cthulhu cycle, and/or [Warhammer's Chaos Gods' minions](https://warhammer40k.fandom.com/wiki/Chaos_Gods) [to include [Malice](https://warhammer40k.fandom.com/wiki/Malice) and [the Horned Rat](https://warhammerfantasy.fandom.com/wiki/Horned_Rat) and the various Skaven ideologies, and lesser-known examples such as [Hashut](https://whfb.lexicanum.com/wiki/Hashut) or [Necoho](https://whfb.lexicanum.com/wiki/Necoho) as [ described by Richard Jones](https://www.quora.com/Should-Warhammer-40k-introduce-factions-of-minor-Chaos-Gods-using-Gods-such-as-Hashut-Necoho/answer/Richard-Jones-138)]). 

To cause or create such beings is at least super-level (I'd argue Principality, but wouldn't be able to prove that). I don't have a good handle on Primordium yet, and I probably won't develop it as a detailed NPC, but I conceive it as being Principality-like (if you were to view Void vs. Chaos and Negent vs. Entropy as diametrically opposed edges of a tetrahedron, then picture Primordium as being like a disc that's radial to Void's and Chaos's fulcrum: not really on the others' axes, yet not exactly orthogonal to them all, just perpetually tangential to the one axis while maintaining a complicated and variable relationship with the other axis – in this fashion, Primordium parallels [Moorcock's Singularity](https://stormbringer.fandom.com/wiki/Singularity) to an extent, insofar as they each wish to cause their respective opposite number's “otherness” to cease existing). That it should expect world-unmaking results from driving some putz like Link crazy indicates either rather poor judgment at the least or vast insight indeed. 

If its power level is that high (more to the point: assuming that it's not _too low_ to consider), then its being off of the game board (so to speak) leaves me wondering as to its nature. Logically its actions must fall within its own purview or that of some [other?] Principality, unless these actions are completely cloaked from their awareness and/or operate at a level above their ability to comprehend. I think that it's safe to assume that the cloaking example isn't the case, but the comprehension level doesn't make sense either. I really don't see Primordium as a Tào-like entity, much less anything on a level with Ayn Sof or Ohr Ayn Sof. 

Perhaps its actions introduce what the [other?] Principalities would view as minor imperfections in the weave, nagging loose threads that must surely derive from their various actions (and perhaps those of Tào), and surely not some agency unknown. 

In any of these cases, you'll note the pronoun “it”. This isn't an issue of uncertainty or coy ambiguity or physical hermaphroditism or fluid identity or PC neutrality or anything: Primordium is sexless and genderless. Some of its actions could potentially be described as masc./fem. in some way or other, but should not be ascribed to masculinity or femininity or whatever-else-inity as such. It simply _is_ an “it” – a living, thinking “it”, but very much an “it”. 

Worth noting is that Primordium's existence lying outside of the Negent “pantheon” implies a multi-dimensionality to the “real” history of Icewall (that is: each _different_ sequence of events and origins could be considered to be true, regardless of the sequences' mutually exclusive natures). As well, Primordium's existence also implies that there could be some sort of thriving ecologically diverse set of “pantheons” at any given tier, not necessarily interactive, but present. 

Similarly, “Malice” (an entirely different one, my own of “ _[The Devil is in the details](https://archiveofourown.org/works/24382933)_ ”) – and the other hosts of such kind – must logically lie within the Sn range or higher, but could arguably be below it. For now, let's just say no more than Sω. 

Why “ _no more than_ ”? Because Malice is perhaps _an_ ultimate expression of iterated power sets, and as such cannot exceed these bounds. Whatever Tào (et alios) might _be_ or be capable _of_ , any upper bounds thereon aren't limited by such bounds as might apply to Malice. 

#####  3: Cosmics' takeaway points 

Laws of nature are malleable to Principalities. Their decision that something operate a certain way is as much a given for them as picking up a glass of water is for most of us (that is: we “choose” to hold it a certain way subconsciously, to move it through a certain path to our mouths, etc. – but we give it hardly any thought at all while a much larger [and still insignificant, though unparsimoniously so] portion of our awareness and being is dedicated to watching TV or replying to an e-mail). They basically _are_ supernatural forces unto themselves. 

Here we can see another clear distinction between individual capabilities and those of the associated civilization. Just as your average human on Earth doesn't wield the same power as an entire nation, the Æsir as a whole wield far greater power than any single Áss. Where any Áss alone might stand at K1 or command K2, the Æsir civilization operates somewhere rather above this (let's say ~K3?) – see the _Icewall_ .pdf technical appendix for a detailed description of the dimensional engineering involved with their star, which is also their computing source. A similar comparison would not be inapt were we to draw such for Kryptonian technologies, though they eschew magic (and psionics). The same could likely be said of the Vanir psionic technologies. 

Spacetime and massenergy are by whim to these beings; civ. >10e26-10e120+ bits data (and this extrapolates only as far as K3 & K4) & ~10e36-10e56 W/y (noting that our observable universe produces something on the order of ~10e45 W). With K4 being the defined maximum within a given universe, I suppose that a being on a multiverse scale (e.g.: Negent, per his own testimony) would presumably weigh in at K5 (minimum). 

Kardashёv Type ~III-V? 

Individuals can be presumed to possess individual powers (sheer ability, though defining this as scientific or magical or zooic [psionics, bio-E, chi, etc.] is probably folly) on the order of Classic TL >27, whether typical or atypical of their civilization. Their civilization overall should be taken as Classic TL 31(+), if there's any difference between individual and collective potentia. 

* * *

####  Fourth tier: a (Super's Super)'s Super **Get to the point ▼ Contents ▲**

Negent referred to other entities or tiers, though he didn't assert that they're definitely real. Tào would presumably qualify for the S↑p _k_ ([Knuth's up-arrow notation](https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation)) tier. Again, just assume that n>1, though in this case “n” is “p”. By way of a quick intro./refresher, if /p/ _here_ refers not to the single-arrow's exponentiation value (after all: just how might we meaningfully raise a non-numeric arrow to an exponent?), but instead to _how many arrows are intended_ (such that S↑2 _k_ = S↑↑ _k_ = _k_ -many “S”s in an iterated exponentiation of S^S^...; a power tower [of S] of the order _k_ ) then: 

2↑04 = {2,2,2,2}  
= 2 + 2 + 2 + 2 = 4 + 4  
= 2 * 4  
= 8 

2↑14 = 2↑2↑2↑2  
= 2 * 2 * 2 * 2 = 2 * 8  
= 22^2 = 24  
= 16 

2↑24 = 2↑↑2↑↑2↑↑2  
= 2 ↑ (2 ↑ (2↑2)) = 2 ^ (2 ^ (2^2))  
= 22^4 = 216  
= 65,536 

2↑34 = 2↑↑↑2↑↑↑2↑↑↑2  
= 2 ↑↑ (2 ↑↑ (2↑↑2)) = 2 ↑↑ (2↑↑4)  
= 2 ↑↑ (2 ^ 22^2) = 2 ↑↑ (2 ^ 24) = 2 ↑↑ (2 ^ 16)  
= 2 ↑↑ 65,536  
= 2^2^2^... (2 w/ 65,535 exponents) 

I'll skip even _trying_ to do 2↑44 and 2↑54\. 

This first table [above] isn't meant to show how quickly the values of 2↑n4 grow, since those are specific to only the number |2|, and the “values” of Sn and Sn aren't exactly numeric things (seriously: how might one really _evaluate_ Sn = S+S+..., or Sn = S*S*..., or S↑2n = S^S^...?). It's meant only to give you a feel of how the up-arrow operator functions. You're familiar with how addition just keeps adding things, and most people can follow the principle of multiplication in general; most people's grasp of things tends to drop off around exponentiation (and even if familiar with something, we all forget things sometimes). That gives you the next operator {∃, I, Δ, +, *, ^, **↑↑** }. 

The next table [below] shows some input- _n_ values across the top with each operator's row giving the respective output-entry values and their intermediate _k_ th-element and iterated sequence steps listed in separate end-columns – I'm skipping some steps, but this isn't meant to be a proof. It's broken into two pieces in an attempt to make it accessible to people reading on their 'phones; for that same reason, the table is truncated to 5 columns, rather than 8 (for that [and some annotating comments], you can find it in the appendix of the “ _I am Legion_ ” .pdf, linked in the summary).  
The numeric results below are irrelevant -- they're _accurate_ for the numbers used **3** , but only meant to demonstrate each operator's rate of change when acting [for the nth time] on the preceding iteration's value. It's not properly constructed, since realistically one should take a given value and apply say... ^2 or ↑↑2, etc. (and n here is really the _i_ th iteration: f(n) at first, and for next n=f(f(n)), and so on), but the results' rates are easier for 2↑↑ than for ↑↑2 (and while not key to general construction of (N)PC power scales for most purposes, combinatorics are pivotal to my own story-universe's nature, hence strongly inclining me toward 2↑↑), so since this is only illustrative, not an actual math treatise, I opted to cut corners here. The point isn't the numbers: it's to parallel the rate of power scaling sketched out in earlier subsections: Tier 1 (2+ = F...A) additive power scale of difference (no superpowers), Tier 2 (2* = S1...5) representing superbeings, etc..  
Don't focus on [the numbers](https://www.dcode.fr/knuth-arrows), just glance at the differences: the difference between n=3 and n=4, for example, for one operator is nothing like the much larger difference for those same _n_ under any later [higher order] operator. That's the point. The {difference between one semi-average person being better or worse at something than the next semi-average person} and the **similar** {difference between a couple of Supers} **isn't** the _same_ difference, not simply in the raw amounts of differences, but even in the differences' sheer ratios; i.e.: not only is {dS4-S3 \- dB-C} > 0 (different in their amounts), but moreover even **{d S4/S3 / dB/C} > 0** itself is _in turn_ a positive slope. 

**oper.** | **n=1** | **n=2** | **n=3** | **n=4** | **n=5** | **. . .**  
---|---|---|---|---|---|---  
**2∃** | 1 | 1 | 1 | 1 | 1 | ...  
**2I** | 1 | 2 | 3 | 2↑-33 = 4 | 5 | ...  
**2 Δ** | 3 | 4 | 5 | 2↑-25 = 6 | 7 | ...  
**2+...** | 2↑-10 = 2+( _i_ n-1=0) = **2** | 2↑-12 = 2+( _i_ n-1=2) = **4** | 2↑-14 = 2+( _i_ n-1=4) = **6** | 2↑-16 = 2+( _i_ n-1=6) = **8** | 2↑-18 = 2+( _i_ n-1=8) = **10** | ...  
**2*...** | 2 | 4 | 8 | 2↑08 = 16 | 32 | ...  
**2^...** | 2 | 4 | 16 | 2↑116 = 65536 | 2↑165536 ≈ 2*1019,729(+) | ...  
**2↑↑...** | 2 | 4 | 65536 | 2↑265536 | 2↑2(2↑265536) | ...  
  
  
  


**oper.** | **element _k_ =** | **iter. seq. =**  
---|---|---  
2∃ | 2↑-3n ≡ 1 | [ _i_ n-1]  
2I | 2↑-2n ≡ n | 1 + _i_ n-1  
**2 Δ** | 2↑-1n ⇒ 2+n | 1 + _i_ n-1  
**2+...** | 2↑0n :: 2*n | 2 + _i_ n-1  
**2*...** | 2↑1n :: 2^n | 2 * _i_ n-1  
**2^...** | 2↑2n = 2↑↑n | 2 ^ _i_ n-1  
**2↑↑...** | 2↑3n | 2 ↑2 _i_ n-1  
  
That second table presents a glimpse of the enormity involved with tier 4 (bottom row, i.e.: ^, or ↑1). The first two rows {2∃, 2I} being my own debatable placeholders (as is 2Δ, really), but in this context each one of those two pretty much boils down to meaning “something exists”: 

  * “2∃” shows only that any given _k_ exists at all (existential operator: the actual value of n being irrelevant, so _k_ =1 in all cases), and 

  * “2I” shows any given entry _k_ to hold the same value as _n_ itself (identity operator: specifically _k_ = _n_ ; the count of a set of existing things being the enumerated iteration of acknowledging the presence of one after another, each given its unique number and so establishing cardinality, but **only marginally iterated in this sense** ). 

  * Its first contextually meaningful row, “2Δ”, is tier 0 (the successor operation: for the successor of _n_ by 2, we find that the value is always _k_ = n+2 [hence the delta “Δ”: the amount of difference between any given n and its _k_ ]; specifically: addition occurs as an operator on n, but it's **not iterated from one result to the next** ), 

  * the second contextually meaningful row, “2+”, is tier 1 (F...A, in which the kth entry holds the value of 21+22+23+...+2n = 2*n; **multiplication: iterated addition** ), 

  * the third contextually meaningful row, “2*”, is tier 2 (Supers: S1...S5: 21*22*23*...*2n = 2^n; **exponentiation: iterated multiplication** ), and 

  * the fourth contextually meaningful row, “2^”, is tier 3 (cosmic beings: S1...S5: 21^22^23^...^2n = 2↑2n = 2↑↑n; **tetration: iterated exponentiation** ). 

  * With the bottom row, “2↑↑”, representing tier 4 (Dreamtime; **pentation: iterated tetration** ), we begin to get a feeling for the sheer scope of power at the 2↑31...5 range when contrasted against the previous tiers' 1...5 ranges' slopes. 

  * Failing a deep grasp of mathematics, I would venture that such a construct might be analogous to treating the usual operators as particular instances of {∃ = op0, I = op1, Δ = op2, + = op3, * = op4, ^ = op5, ↑↑ = op6} – though subscripts (re-indexed to 1...7?) might be more appropriate than superscripts – but that's too far astray from our purposes here. 




This isn't a perfect analogy, of course, just a rough set of scales per operator. It shows how any given S↑p _k_ 5 is vastly beyond any lower-p tiers' 5th elements {7, 10, 32, 2*1019,729(+), 2↑2(2↑265536)}, which gives us some sense of scale – but it also leaves most of their initial elements equal {3, 2, 2, 2, 2}, and that (i.e.: F ≈ S1 = S1 = S↑31) just isn't the case. In this, perhaps we should view them as { _k_ th element} ordinals rather than as |n = value| cardinals. 

Regardless of the specifics above, and whether Tào et al. are real (within my story world's hierarchy), if we follow such a ranking structure, and we rank Tào someplace within this tier, then Tào would be of Rank S↑31...5. 

Following this layout, we can use the up-arrow notation for the other tiers right along with Tào as follows: 

  * 0 (“ε”, e.g.: paramecia) :: Rank S↑-11...5; 

  * 1 (average F...A) ≡ Rank S↑01...5; 

  * 2 (Supers and Powers) = Rank S↑11...5; 

  * 3 (Principalities: Void, Chaos, Entropy, and Negent – and the putative Other) = Rank S↑21...5; 

  * **4 (Tào) = Rank S↑** 3 **1...5;**

  * _5 (Ayn Sof, Ohr Ayn Sof – more on them in next entry down) ⇒ ≥Rank S↑ 41...5._




#####  4: Dreamtime takeaway points 

Law and nature don't really apply here anymore. At this point, it's all Dreamtime manifested. Every _thing_ implies _all_ other things, so that it's more a question of “where” you are in the plenum than of what “is” or “isn't”. To “be” is to dream, to dream is to “perceive”, to “perceive” is to manifest as being. 

One might conclude that if a being's _civilization's_ K type is arguably one type greater than a being's **own** , then Principality (= tier three) civilization (:: tier four?) might well stand at this Dreamtime level. 

“Reality” and “unreality” are a meaningless distinction to someone at this tier; even “individuals” (aspects?) :: ≫10e120 bits data & :: >10e66 W/y, insofar as {bits, data, watts, power, or a year} can be said to have any meaning here either. 

Kardashёv Type ~VI. 

Traveller TL loses meaning by this stage (and might very well have done so before even tier three), but take beings at tier four as Classic TL ≫31. 

* * *

####  Fifth tier: to infinity and beyond **Contents ▲**

Ohr Ayn Sof (and even Ayn Sof alone) would normally be taken as infinite, though in the above tier system one could suppose another hyperoperator beyond the up-arrow, but it's more of a notation than an actual operator (then again, aren't they all?). In this sense, the fifth tier would be a function of {element _k_ = S↑41...5 ⇒ sequence = S↑3 _i_ n-1} and Kardashёv Type >VI. 

Aside from the values of _k_ (vast) and the implications of K type VII+ equivalent power / knowledge / perception / etc. (meaningless to me: absolutely aware of everything at once), it's probably safe to assume that these don't actually apply to them. Perhaps these _k_ values apply to someone in the Icewall food chain, but I'm pretty sure that Ayn Sof and Ohr Ayn Sof are the source of all. Period. Full stop. 

Do I know this? Well, no. Yes, I'm the guy writing it all, but I haven't designed that particular aspect as yet, and I doubt that I ever will. 

I suppose that as a tier _in itself_ – whether it applies to any (N)PC or not – tier 5 would simply follow the pattern set forth so far: 5 = S↑4n, hence having (N)PCs of Rank S↑41...5. 

You want to know where Ayn Sof should _logically_ fall? <sigh> Let's say that it would have to be something like tier Ω. 

* * *

####  TL;DR: **Contents ▲**

So, where does all of that leave us? It leaves us contemplating a universe from the perspective of a grain of sand. That's not a statement of insignificance, either – it's multilayered: reality as a fractal, logarithmic scales of scientia and potentia, and able to view at least one tier upward and downward from a given position. 

Each tier might see the downward tiers as being like fragile soap bubbles, or mere shadowplay, or ephemeral figments of thought. Looking upward might be akin to a caveman trying to grasp the workings of a castle, or a Death Star, or the inner mind of Cthulhu. 

For simplicity, assume further that within a given tier, someone might personally possess different specific ranks of knowledge vs. perception vs. power vs. etc., _command_ some higher rank of each (such as personal possessions, legal authority, etc.), and yet **be part of a civilization** that is perhaps at least one full rank higher still (all of this leading to upward of an entire K-typal difference between the individual and the world that they live in). In this way, the range covered by a rank R within a given tier T might be considered to be equivalent to R = 0.20 ΔT; it's a bit awkward, since although the ranks of each tier progress smoothly, successive tiers progress with the next higher derivative: ΔT1 < ΔT2 < etc. (i.e.: a smooth slope per tier, but with each tier's slope being greater than the previous one's). 

**Tier** | **Population** | **Ranks** | **Individual capabilities** | **Tech level**  
---|---|---|---|---  
**0** | Paramecia, etc. | N/A | Kardashёv Type ε | TL ε  
**1** | Human beings | F(-)...A(+) | Kardashёv Type 0 | CTL 6-9  
**2** | Supers & Powers | S1...S5 | Kardashёv Type ~I-II? | CTL 9-20  
**3** | Principalities | S1...S5 | Kardashёv Type ~III-V+? | CTL >27-31+  
**4** | Tào(?) | S↑31...S↑35 | Kardashёv Type ~VI | CTL ≫31  
**5 / ω** | Ayn Sof(?) | S↑41...S↑45 | Kardashёv Type VII+ | \--  
  
**N.B.:**

If the mathematics grabbed you more than the general idea did, and you feel a need to run it into the ground, then you might want to look into things like the [Ackermann](https://www.youtube.com/watch?v=CUbDmWIFYzo) [function](https://www.youtube.com/watch?v=i7sm9dzFtEI), Graham's number, Loader's number, the TREE function (note: TREE(n) ≠ tree(n)), and [fast growing hierarchies](https://en.wikipedia.org/wiki/Fast-growing_hierarchy#Functions_in_fast-growing_hierarchies).  
For a quick intro. to these, try these videos: 

  * [Graham's Number – Numberphile](https://www.youtube.com/watch?v=XTeJ64KD5cg&list=PLt5AfwLFPxWJ_FwchNAjB3xa8UnoKdmQI&index=24) (explaining the basic nature of the question that Graham's number answers), 

  * [What is Graham's Number? (feat. Ron Graham)](https://www.youtube.com/watch?v=HX8bihEe3nA&list=PLt5AfwLFPxWJ_FwchNAjB3xa8UnoKdmQI&index=16) (explaining the n-cubic question itself), 

  * [Graham's Number Escalates Quickly – Numberphile](https://www.youtube.com/watch?v=txajrEOTkuY&list=PLt5AfwLFPxWJ_FwchNAjB3xa8UnoKdmQI&index=14) (expanding on the process to reach it, fleshing out just how insanely large a number we're looking at), 

  * [The Enormous TREE(3) – Numberphile](https://www.youtube.com/watch?v=3P6DWAwwViU) (showing just what the TREE game is), 

  * [TREE(Graham's Number) (extra) – Numberphile](https://www.youtube.com/watch?v=wqdOnEnfJXM) (this one in particular looks at the Shannon data entropic limits of the universe), 

  * [TREE vs Graham's Number – Numberphile](https://www.youtube.com/watch?v=0X9DYRLmTNY) (expanding on the iteration of f(f(f(...(f(n))...) and introducing {ε0 = ω↑2ω, ε1, ε2, …, εε1, …, ζ0 and η0 via Veblen hierarchies of functions, and eventually the Feferman–Schütte ordinal Γ0}). 




**O ~~~ O**

**Author's Note:**

>  **1 ▲** Subscribers might recognize the slight but crucial difference between “cuccos” and “Cuccos” there, but for first-timers, the key here is the capitalization: aside from normal capitalization rules of English, I add one further stipulation that the proper name of any intelligent species is capitalized in the same manner as English capitalizes nationalities (like Tolkien, though usually more invariant).  
>   
> This is a variant (in the Icewall meta-setting) of the terminology [used by Orion's Arm](https://www.orionsarm.com/eg-topic/45b1774e4ba77) (deviating regarding sophonce):
> 
>   * Sapient = problem solving, like a calculator or a slime mold;
>   * Sentient = self-aware, knowing that one exists – that there is an “I” in the mirror, observing one's own thoughts;
>   * Sophont = other-aware, feeling for others' plights and suffering (empathy in modeling “other”, not ESP).
> 

> 
>   
> **2 ▲** Mathematically, ZF(C) refers to different infinite sets being equal or unequal. The cardinal numbers (being the same size as the set of rational numbers) are ℵ0, arguably the “smallest” infinity; ordinals are ℵ1 (infinitely “more” of them); geometric curves ℵ2. The real numbers' position (coinc. = rationals' power set size 2ℵ0) is debatable, hence referred to as ℶ1; discontinuous functions of real numbers are likewise ℶ2.
> 
> For more info, see {Wikipedia, Wolfram MathWorld, DavidDarlingInfo, StackExchange, Quora, etc.} re. Cantor's diagonal slash, aleph {null, one, two}, beth numbers, (un)countable infinities, ZF(C), and (G)CH.
> 
> ZF(C) plays a crucial role in my combined story-world's future that no longer will have happened. In an extremely truncated thumbnail: civilization reached truly epic levels, discovered the reality-determining morphogenic effects of Zermelo-Fraenkel (with/without Choice) and CH, and became embroiled in a “war” (had some unspecified portion of beings come to align philosophically with infinite sets being meaningfully different (or not) in “size” [sort of], then the wave function of reality would have collapsed around that single arrangement, negating all of the “tech” [sort of] that relied on the other (aleph/beth Encyclopædiae relied on Banach-Tarsky). Seeking to avoid this, a third faction hit the button and exploded spacetime, all spatiotemporal subsets disappearing instantly, replaced with a new [monadic-multiverse-in-one-ℶ1-place] ℶ2-history, and the jigsaw pieces falling back everywhen to **interweave** into it higgledy-piggledy. Complicating matters is that there are some things in the setting that work under either set of axioms, others that don't rely on either set at all, some that must have both (contradictorily), and still others that require the _option_ of either set.
> 
>  **Cf.:** “[EVA](https://archiveofourown.org/works/24363373)”, [“Meat Pies” (ch. 3's footnote 3](https://archiveofourown.org/works/24362857/chapters/60135520#3) re. Cor, w.r.t. the overall story's setting), “[Megamechs 101](https://archiveofourown.org/works/24385531)”, and the “[Tell me more about this Earth-thing](https://archiveofourown.org/series/1975186)” series re. Kardashёv tech and post-level civilizations (esp. in light of the anime “Fairy Tail” and “One Piece”, or Gregory Benford's “Galactic Center” novel series), following the events of “[Nothing's gonna change my world](https://archiveofourown.org/works/24380977)”.  
>   
>   
>  **3 ▲** For the OCD: the numbers in the first two rows are the consequence of treating the later ↑ operators as differentiable; those rows' cell and sequence rules in the second half of the table come from each operator's element rule being the next hyperoperator's iteration rule.  
>  The result is a zig-zag of each operator's element- _k_ rule being the **next** hyperoperator's iteration rule to be performed on each _k_ n to get the next _k_ n+1. Taking that and working with the known zig-zag of rules, we can extend the pattern backward to lower-order “hypo”-operators (as it were), and obtain things like Δ _k_ n=2+n, which in turn tells us immediately that the iterating rule for Δ follows as 1+ _k_ n-1.
> 
>  **Bonus: ↑** If you got this far, then you've probably waded through a fair amount of data. As a reward, here's a tutorial for further study: “[Fonts, and colors, and work skins, oh my!](https://archiveofourown.org/works/28934610)”


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